Evaluation of the QoS policy model of an ordinary 5G smart city cluster with predominant URLLC and eMBB traffic

A typical element of the smart city’s information and communication space is a 5G cluster, which is focused on serving both new and handover requests because it is an open system. In an ordinary 5G smart city cluster, Ultra-Reliable Low-Latency Communications (URLLC) and enhanced Mobile BroadBand (eMBB) traffic types prevail. The formation of an effective QoS policy for such an object (taking into account the potentially active slicing technology) is an urgent problem. As a baseline, this research considers a Quality of Service (QoS) policy with constraints for context-defined URLLC and eMBB classes of incoming requests. Evaluating the QoS policy instance defined within the framework of the basic concept requires the formalization of both a complete qualitative metric and a computationally efficient mathematical apparatus for its calculation. The article presents accurate and approximate methods of calculating such quality parameters as the probability of loss of typed requests and the utilization ratio of the communication resource, which depend on the implementation of the estimated QoS policy. At the same time, the original parametric space includes both fixed characteristics (amount of available communication resources, load according to request classes) and controlled characteristics due to the specifics of the implementation of the basic QoS concept. The paper empirically proves the adequacy of the presented mathematical apparatus for evaluating the QoS policy defined within the scope of the research. Also, in the proposed qualitative metric, a comparison of the author’s concept with a parametrically close analogue (the well-known QoS policy scheme, which takes into account the phenomenon of reservation of communication resources), determined taking into account the reservation of communication resources, was made. The results of the comparison testify in favour of the superiority of the author’s approach in the proposed metrics.


Introduction and state-of-the-art
The creation of the 5G platform was a response to the growing need to solve problems such as [1][2][3]: growth in mobile traffic, an increase in wireless network-connected devices number, and a need to reduce delays when introducing new services.The main means to solve this impressive list of problems are three leading technologies [4,5]: Ultra-Reliable Low-Latency Communications (URLLC), enhanced Mobile BroadBand (eMBB), and massive Machine-Type Communications (mMTC).The mMTC technology is focused on serving the traffic of a huge number of applications and devices that do not demand latency.The URLLC technology is designed to support mission-critical communications with low latency, high security and reliability.Finally, eMBB technology aims to provide high data transfer rates.This issue is discussed in more detail in [2].
The most relevant platform for the real application of all the capabilities of the 5G platform is the information space of a smart city.To a first approximation, this space can be segmented into industrial, specialized and civil clusters that require wireless communication coverage.At the same time, it is civic clusters that dominate the information and communication landscape of the modern agglomeration.This circumstance allows us to further call the "civilian" 5G cluster ordinary.At the same time, typical information exchange in such a 5G cluster requires the implementation of URLLC (ensuring the information needs of critical infrastructure and institutions) and eMBB (ensuring the information needs of citizens and the service sector) technologies [6].In this context, the urgent problem arises of providing high-quality service to the needs of users of an ordinary 5G cluster while simultaneously using URLLC and eMBB technologies.
Aspects of the joint use of URLLC and eMBB technologies are being studied by many teams of scientists.This is confirmed by current review articles [7][8][9][10][11] and the impressive list of specialized studies that are mentioned in them.The authors of articles [12,13] propose to allocate resources for URLLC traffic, prioritizing the reliability of connections.In [12,14,15], a possible architecture for joint transmission of URLLC and eMBB types of traffic was explored.The works [16,17] discuss the main scenarios for the implementation of URLLC and eMBB types of traffic service, technical requirements, as well as features of non-orthogonal resource sharing provided that eMBB, mMTC and URLLC consumers are located in the network.Articles [17][18][19] present the results of the study of the effectiveness of various modulation options for orthogonal frequency division multiplexing in the parametric space of their spectral efficiency, reliability, etc.The works [20,21] consider options for non-orthogonal use of resources to support URLLC and eMBB connections.Among the relevant studies, we highlight the approach [22] where resource reservation is used to improve the Quality of Service for URLLC and eMBB types of traffic.In [23][24][25], the application of Network Slicing technology in the case of heterogeneous multiple access (both orthogonal and non-orthogonal) was studied.In addition to URLLC priority access, the authors of articles [26,27] focus on maintaining the QoS of eMBB traffic, using methods of stochastic geometry and queuing theory.
In all the works mentioned above, the authors either view the 5G cluster as a closed system or do not take into account the features of the simultaneous operation of URLLC and eMBB technologies on the 5G platform.At the same time, the closest analogues to this research are works [28,29].A feature of the works [28,29] is the use of queuing systems with random requirements for modelling URLLC-eMBB interactions.At the same time, we note that the authors of the mentioned studies try to abstract from the conditions in which the 5G cluster will operate.Thus, while increasing the universality of the results obtained, the proposed solutions lose adequacy for specific objects.We should also pay attention to the fact that the authors formulate their models [26][27][28] in a well-looking, but cumbersome mathematical basis, which does not allow the methods obtained to be used to analyze real 5G clusters with a large amount of communication resources.In this article, we focus on addressing these limitations.
Next, we summarize the main attributes of our research in a compact form.
The object of the study is the process of managing the quality of service of typified incoming requests in the open 5G ecosystem of a smart city cluster with predominant URLLC and eMBB traffic.
The subject of research includes methods of probability theory and mathematical statistics, recovery theory, Markov process theory, and experiment planning theory.
The aim of the research is formulated as follows: to create a computationally efficient, holistic concept for evaluating an arbitrary instance of the QoS policy of an ordinary 5G smart city cluster with predominant URLLC and eMBB traffic, which will make it possible to close the research gap identified as a result of the analysis of analogues.
Research objectives are: • to determine the parametric space of stable and controlled characteristics that characterize the research object (presented in Section 2.1), • to formulate a comprehensive qualitative metric for evaluating an arbitrary instance of the implementation of the research object (presented in Section 2.2), • to formulate an accurate parameterized concept for evaluating an arbitrary instance of the QoS policy of an ordinary 5G smart city cluster (presented in Section 2.3), • to formulate a computationally efficient approximate parameterized concept for evaluating an arbitrary instance of the QoS policy of an ordinary 5G smart city cluster (presented in Section 2.3), • to analytically investigate partial cases of QoS policy implementation of an ordinary 5G smart city cluster in terms of an approximate evaluation concept (presented in Section 2.4), • justify the adequacy of a mathematical apparatus proposed and demonstrate its functionality in the context of a research aim (presented in Section 3 and discussed in Section 4).
We define the main contribution of the research as follows.As a baseline, this research considers a QoS policy with constraints for context-defined URLLC and eMBB classes of incoming requests.Evaluating the QoS policy instance defined within the framework of the basic concept requires the formalization of both a complete qualitative metric and a computationally efficient mathematical apparatus for its calculation.The article presents accurate and approximate methods of calculating such quality parameters as the probability of loss of typed requests and the utilization ratio of the communication resource, which depend on the implementation of the estimated QoS policy.At the same time, the original parametric space includes both fixed characteristics (amount of available communication resources, load according to request classes) and controlled characteristics due to the specifics of the implementation of the basic QoS concept.
The article is organized as follows.Section 2 presents the main scientific result of the study.In particular, the research was set up, including the formation of the main parametric space of the model of the studied process.Next, a qualitative metric is formulated for evaluating an arbitrary instance of the studied process.Exact and approximate methods are formulated for the evaluation of the QoS policy model of an ordinary 5G smart city cluster (including some typical cases).In Section 3, from the standpoint of mathematical statistics and the experiment planning theory, the results of evaluating an instance of a QoS policy model of an ordinary 5G smart city cluster are presented, both using the author's approach and in comparison with an analogue.Section 4 analyzes the obtained empirical results.Section 5 draws general conclusions on the work and formulates directions for promising further research.

Statement of the research
The focus of our study was the process of supporting 5G cluster application groups with different quality of service (QoS) requirements.Focusing on the 5G platform, three such groups are distinguished: URLLC, eMBB and mMTC.An ordinary 5G smart city cluster mainly deals with applications that belong to the first two groups (mMTC group applications are typical of industrially oriented 5G clusters).Accordingly, for an ordinary 5G cluster, an actual problem is the evaluation of QoS policies aimed at supporting URLLC and eMBB applications.
Suppose that the channel resource of an ordinary 5G cluster includes C2N units, which are oriented to support Poisson flows of: • a new incoming requests of URLLC type with intensity η u ; • a new incoming requests of eMBB type with intensity η e ; • a handover of incoming requests of URLLC type with intensity η hu ; • a handover of incoming requests of eMBB type with intensity η he ; One channel resource unit is sufficient to support a URLLC request, while a�C channel resource units are required to support an eMBB request.To simplify the model, let's resort to the admissible assumption that eMBB requests are inelastic (at the moment of completion of the service of such a request, all resource units allocated for its support are released at the same time).
All functions that characterize the distribution of time spent on supporting {u,e,hu,he}requests belong to the exponential type but with different means, which are determined by parameters {μ u ,μ e ,μ hu ,μ he }, respectively.
Let's define the QoS policy considering that the class of URLLC-requests has a higher priority than a class of eMBB-requests, and within the corresponding class, handover requests have a higher priority than new ones.We will ensure the balance of the QoS policy under the following conditions: • the number of accepted u-requests cannot exceed the threshold T u , • if, upon receipt of a hu-request, the base station has at least one free resource unit at its disposal, it is directed to support this request, • the number of accepted e-requests cannot exceed the threshold T e , • the number of accepted he-requests cannot exceed the threshold T he , • 0<T e �T he �T u �C.
We will consider the ratio of channel resource utilization � C and the probability of loss of requests of each of the defined types: {P u ,P hu ,P e ,P he } as an analytical metric of QoS policy quality.

Accurate estimation of the QoS policy model of an ordinary 5G smart city cluster
Suppose that a = 1.In the context of a research statement, at an arbitrary moment, the state of an ordinary 5G cluster is characterized by the vector !n ¼ ðn Su ; n Se Þ, where n Su = n u +n hu a total number of received URLLC requests and n Se = n e +n he a total number of received eMBB requests, respectively.A phase space of a Markov chain, which corresponds to the two-dimensional vector !n, is described by the expression Note that an inscription of the type y≔x is interpreted so that the value of y is determined by the expression x.
Taking into account the conditions for defining the QoS policy formulated in Section 2.1, elements of a Markov chain (1) generating matrix will be determined from the following ratios: where The analytical representation of QoS conditions embodied in expression ( 2) is determined through a stationary probability distribution of a process (1) states.Assume that this is a stationary probability of losing requests of a type r, r2{u,e,hu,he}.According to the PASTA theorem [30], these parameters are defined as a corresponding marginal distribution of a Markov chain (1), i.e.: pð! nÞdðn Su þ n Se ; CÞ; ð4Þ where pð! nÞ is a state !n 2 F stationary probability; δ(�) is the Kronecker delta; I(E) is an event E indicator function.Based on this terminology, the average number of used communication resources of a 5G cluster can be defined as Based on the Kolmogorov-Arnold theorem [31], it can be stated that the two-dimensional Markov chain (1) is characterized by reversibility.That is, in the process F there is zero circulation between the states (a local balance condition is fulfilled).In turn, this allows expressing a state (i,j) probability in terms of a state (0,0) probability by choosing an arbitrary route between these two states in a corresponding graph.Therefore, in a case T he +T u �C of the route (0,0), (1,0), . .., (i,0), (i,1), . .., (i,j), the multiplicative solution for a stationary probability distribution pð! nÞ of a process F is defined by an expression where w Se ≔Z Se =m Se , w Su ≔Z Su =m Su , w he ≔Z he =m Se , w hu ≔Z hu =m Su ; p(0,0) is determined as a result of the normalization of where If T he +T u >C, then a states stationary probability distribution of a process F will take a form where p(0,0) is determined as a result of pð0; 0Þ ¼ 1= where With a stationary probability distribution of target process F states calculated according to expressions (8) or (9), the desired quality indicators of the formed QoS policy are calculated according to expressions (3)- (7).
The proposed approach to evaluating the QoS policy of an ordinary 5G smart city cluster is accurate.The use of this term is correct, because to calculate pð! nÞ, !n 2 F, with the help of a mathematical apparatus presented in this section, researchers will have to generate the entire phase space of the states of the evaluated model.At the same time, for a large value of C, researchers will have difficulties with the calculations of expressions ( 8) or (9), because they will have to calculate a large number factorials and also raise to a power either values close to zero (at a low load on the 5G cluster) or large values (when the 5G cluster is heavily loaded).This circumstance prompted the authors to formalize the concept of the approximate calculation of the metric (3)- (7), which is presented in the following subsections of Section 2.

Approximate evaluation of the QoS policy model of an ordinary 5G smart city cluster
Let's formulate the concept of approximate calculation of the metric (3)-( 7), for the implementation of which it will not be necessary to generate an entire estimated QoS model states phase space.An initial postulate for the concept explained below is that tabulated values are used in expressions (8), and (9).With a large difference between the values of the parameters of different types of traffic, this circumstance will allow us to achieve the desired simplification of the calculation of stationary probabilities of the states of the evaluated model with high accuracy.
First, let's assume that for the evaluated model, URLLC traffic characteristics prevail over eMBB traffic characteristics: μ Su >>μ Se .At the same time, we take into account that URLLC connections are short-lived compared to eMBB connections [2], however, the share of active URLLC connections in the traffic of the studied 5G cluster is dominant.
Let's divide the state graph of a investigated model by a value of a state vector's first component: For the classes formed as a result of the implementation of partition (10) to an original phase space of a model F states, it is characteristic that a transitions probabilities between states belonging to a same class significantly exceed a transitions probabilities between states belonging to different classes.We cluster a states F i classes into the integral state hii.The corresponding clustering function is described by an expression Kð! nÞhii8!n 2 F i , i ¼ � 0; T he .Such a clustering function defines an integral model, which is described by a one-dimensional Markov chain with state space F≔fhii : i ¼ � 0; T he g.Accordingly, the states stationary probability distribution is defined as pði; jÞ � a i ðjÞbðhiiÞ; i ¼ � 0; where fa i ðjÞ : ði; jÞ 2 F i g is the stationary probability distribution of a states within the class S i , and where fbðhiiÞ : hii 2 Fg is a stationary probability distribution of an integral model F states.When determining these stationary state probability distributions, we will consider such cases as h1 : We denote a model F phase matrix elements with a space of states F i as q i (j,l).For the case h1, generating matrix elements are defined in the same way: After analyzing expression (12), we conclude that a stationary probability distribution of a model F states with a state space F i is identical to the stationary probability distribution of an Erlang model of type M/M/C−i/0 [32] states with state-dependent intensities of an arrival of new requests and an intensity μ Su of serving received requests by a single device.Based on this conclusion, we write: where Having generalized expressions ( 14) and (2), we formalize the relationship for calculating the generating matrix qðhii; hi 0 iÞ, hii; hi 0 i 2 F: qðhii; hi 0 iÞ ¼ The relations defined by expression (15) will allow us to determine a stationary probability distribution of a model F states (a one-dimensional birth and death Markov process): Finally, having determined the characteristics ( 13)-( 17), we formalize the expressions for the approximate estimation of the metric (3)- (7) for the evaluated instance of the QoS policy model at μ Su >>μ Se : bðhiiÞa i ðC À iÞ; ð19Þ ( For a case h2, a stationary probability distribution of a model F states with state space S i , i ¼ � 0; C À T u À 1, is also determined by relation (15), and a stationary probability distribution of a model F states with state space S i , i ¼ � C À T u ; T he , is identical to a stationary probability distribution of an Erlang model M/M/C−i/0 states with load w Su .Accordingly, expressions ( 19)-( 22) obtained for a case h1 will remain adequate for the case h2.The qualitative indicator P u for the case h2 is characterized by the expression

Approximate evaluation of typical QoS policy models of an ordinary 5G smart city cluster
Let's consider several typical implementations of the QoS policy of an ordinary 5G smart city cluster.
We formalize the evaluation procedure of the fully available QoS policy, the characteristic feature of which is to ignore the differences between URLLC and eMBB traffic types: T e = T he = T u = C (this corresponds to the case h2 described in the previous subsection).It is obvious that in such a case P u = P hu = P e = P he = P. Considering the PASTA theorem, let us generalize expressions ( 19)-( 23) for the following partial case: where EB(x,y) is the classical expression for an Erlang model M/M/y/0 (Erlang B Model) at load x.
The stationary distribution (15) for this partial case is defined as a stationary distribution of a model M/M/C−i/0 states probabilities with load w Su : qðhii; hi 0 iÞ ¼ Now we formalize the QoS evaluation procedure of a single-threshold policy, the characteristic feature of which is to ignore discrepancies between new and handover requests for both URLLC and eMBB traffic types: T e = T he , T u = C, T he <T u (this corresponds to the case h2).Based on expressions ( 19)-( 23) for such a partial case, we obtain: The qualitative parameter � C for this partial case is calculated by expression (22).Accordingly, we present the relation (15) as follows: qðhii; hi 0 iÞ ¼ Finally, we formalize the QoS policy evaluation procedure for a 5G cluster, where eMBB traffic parameters dominate over URLLC traffic parameters: η Su <<η Se , μ Su <<μ Se .
For such a partial case, an initial states F phase space is divided by a value of a state vector's second component: Using the semantically identical statements presented in the previous subsection, we formulate the information technology for determining the qualitative metric (3)-( 7) for the following partial case: Stage 1.We calculate a stationary probability distributions of a model F states with a state space F i , i ¼ � 0; C. Such distributions: • -for a model with a state space F i , i ¼ � 0; C À T he , coincide with a probability distributions of a model M/M/T he /0 states; • -for a model with a state space At the same time, in all models, an intensity of a unit of communication resource service is equal to μ Se , and an intensity of a flow of incoming requests η l depends on a state of l: Z he 8l � T e : ( Stage 2. If T he +T u �C, then an elements of an integral model generating matrix are determined by the ratio qðhii; hi 0 iÞ ¼ If the inequality T he +T u �C is not fulfilled, then we proceed to Stage 3, otherwise, we proceed to Stage 4.
Stage 3. Elements of an integral model generating matrix are determined by a ratio qðhii; hi 0 iÞ ¼ Stage 4. Similarly to (16), and (17), we find a stationary probability distributions of a model β(hii) states, i ¼ � 0; C, and, after determining the stationary probability distribution of a model F by expression (11), we calculate an approximate value of an evaluated QoS policy quality indicators.In this case: • if the inequality T he +T u �C is satisfied, then the approximate values of the quality parameters (3)-( 7) for the evaluated QoS policy are calculated using the expressions bðhiiÞa i ðC À iÞ; ð27Þ • an incoming u-request is accepted if, at a moment of its receipt, a number of occupied units of the channel resource is less than N u ; • an incoming e-request is accepted if, at a moment of its receipt, a number of occupied units of the channel resource is less than N e ; • an incoming he-request is accepted if, at a moment of its receipt, a number of occupied units of the channel resource is less than N he .
In both the author's model QoS A and the competing model QoS C , the total number of channel resource units C is constant, and the parameters hT u ,T he ,T e i (for QoS A ) and parameters hN u ,N he ,N e i (for QoS C ) are controllable.
Of research interest is the comparison of policies QoS A and QoS C in the qualitative metric r.The rest of the input data for the experiment are defined by the set QoS 1 (see the beginning of this section).The results of the comparison of the competing QoS policy with reservation QoS C and the author's QoS policy QoS A are presented in Figs 6-9.

Discussion
Let's start a discussion by analyzing a simulation results, which are shown in Figs 1-3.As already emphasized, the purpose of this experiment was to prove an adequacy of the concept AM presented in Section 2.3 for the approximate evaluation of an arbitrary instance of the QoS policy model of an ordinary 5G smart city cluster, represented in the parametric space of the initial characteristics defined in Section 2.1.The concept EM for accurate estimation of a  Finally, let's comment on the presented in Figs 6-9 results of comparison of the competing QoS policy with reservation QoS C and the author's QoS policy QoS A for the 5G cluster, the initial parameters of which were set by the set QoS 1 .From Figs 7 and 8, it can be seen that the graphs that correspond to the qualitative indicators P u , P e , P he for the model QoS A are superior to the competitor QoS C , while the graphs that correspond to the qualitative indicator � C (see Fig 9) for these competing models practically coincide.Therefore, the model QoS C is superior to the author's model only in terms of quality indicators P hu .Taking into account that the complexity of the implementation of both models QoS C and QoS A is the same, it can be reasonably argued that for a 5G cluster, the initial characteristics of which are caused by the set QoS 1 , the author's model outperformed the competitor.However, we note the locality of this conclusion, because for a different set of values of the initial characteristics, the evaluation results of these QoS models may differ significantly, which requires a separate investigation.

Conclusions
A typical element of the smart city's information and communication space is a 5G cluster, which is focused on serving both new and handover requests because it is an open system.In an ordinary 5G smart city cluster, URLLC and eMBB traffic types prevail.The formation of an effective QoS policy for such an object (taking into account the potentially active slicing technology) is an urgent task.As a baseline, this research considers a QoS policy with constraints for context-defined URLLC and eMBB classes of incoming requests.Evaluating the QoS policy instance defined within the framework of the basic concept requires the formalization of both a complete qualitative metric and a computationally efficient mathematical apparatus for its calculation.The article presents accurate and approximate methods of calculating such quality parameters as the probability of loss of typed requests and the utilization ratio of the communication resource, which depend on the implementation of the estimated QoS policy.At the same time, the original parametric space includes both fixed characteristics (amount of available communication resources, load according to request classes) and controlled characteristics due to the specifics of the implementation of the basic QoS concept.
The paper empirically proves the adequacy of the presented mathematical apparatus for evaluating the QoS policy defined within the scope of the research.Also, in the proposed qualitative metric, a comparison of the author's concept with a parametrically close analogue, determined taking into account the reservation of communication resources, was made.The results of the comparison testify in favour of the superiority of the author's approach.
Note that the presented mathematical apparatus has the potential to expand the range of QoS policies admissible for evaluation.The field of potentially acceptable objects includes network technologies that provide for the formation of queues for incoming requests.The formalization of relevant analytical adaptations of the mathematical apparatus presented in the article is a promising direction for further research.
Also note that we presented the main scientific result of the study based on mathematical statistics, and recovery theory.This approach is characterized by sufficient universality (subject to adequacy control) and low computational complexity.The shortcoming of the proposed approach is that the procedure for adapting the proposed mathematical apparatus to the specific conditions of 5G network deployment cannot be automated.In this context, a promising direction for further research is also the synthesis of models based on artificial intelligence, which can complement the result presented in the article.The authors see works such as [33][34][35][36][37] as analogues for this area of research.

Fig 8 .
Fig 8. Comparison of QoS C and QoS A policies in the parametric space of quality indicators P e , P he .https://doi.org/10.1371/journal.pone.0295252.g008

Fig 9 .
Fig 9. Comparison of QoS C and QoS A policies in the parametric space of quality indicator � C � .https://doi.org/10.1371/journal.pone.0295252.g009 Se � T he ; n Su � T u ; e � n